How to go about classifying 1000 classes?
If you are asking for arbitrary ML task dealing with 1000+ classes the most straigtforward thing that comes to mind is the ImageNet - https://en.wikipedia.org/wiki/ImageNet#cite_note-nytimes_2012-2. It has more the 20k categories in the present time.
In order to perform such classification task you need

A large enough dataset, such that each class occurs frequently enough or efficient data augmentation
A powerful model, such that can adapt to variety of different problems

The first neural network to tackle this problem was AlexNet and since then plethora of architectures : VGG's, ResNet's, EfficientNet got better and better quality.
They have a lot of filters that react on different patterns.